1 /*-
2 * SPDX-License-Identifier: BSD-3-Clause
3 *
4 * Copyright (c) 1992, 1993
5 * The Regents of the University of California. All rights reserved.
6 *
7 * Redistribution and use in source and binary forms, with or without
8 * modification, are permitted provided that the following conditions
9 * are met:
10 * 1. Redistributions of source code must retain the above copyright
11 * notice, this list of conditions and the following disclaimer.
12 * 2. Redistributions in binary form must reproduce the above copyright
13 * notice, this list of conditions and the following disclaimer in the
14 * documentation and/or other materials provided with the distribution.
15 * 3. Neither the name of the University nor the names of its contributors
16 * may be used to endorse or promote products derived from this software
17 * without specific prior written permission.
18 *
19 * THIS SOFTWARE IS PROVIDED BY THE REGENTS AND CONTRIBUTORS ``AS IS'' AND
20 * ANY EXPRESS OR IMPLIED WARRANTIES, INCLUDING, BUT NOT LIMITED TO, THE
21 * IMPLIED WARRANTIES OF MERCHANTABILITY AND FITNESS FOR A PARTICULAR PURPOSE
22 * ARE DISCLAIMED. IN NO EVENT SHALL THE REGENTS OR CONTRIBUTORS BE LIABLE
23 * FOR ANY DIRECT, INDIRECT, INCIDENTAL, SPECIAL, EXEMPLARY, OR CONSEQUENTIAL
24 * DAMAGES (INCLUDING, BUT NOT LIMITED TO, PROCUREMENT OF SUBSTITUTE GOODS
25 * OR SERVICES; LOSS OF USE, DATA, OR PROFITS; OR BUSINESS INTERRUPTION)
26 * HOWEVER CAUSED AND ON ANY THEORY OF LIABILITY, WHETHER IN CONTRACT, STRICT
27 * LIABILITY, OR TORT (INCLUDING NEGLIGENCE OR OTHERWISE) ARISING IN ANY WAY
28 * OUT OF THE USE OF THIS SOFTWARE, EVEN IF ADVISED OF THE POSSIBILITY OF
29 * SUCH DAMAGE.
30 */
31
32 /*
33 * The original code, FreeBSD's old svn r93211, contained the following
34 * attribution:
35 *
36 * This code by P. McIlroy, Oct 1992;
37 *
38 * The financial support of UUNET Communications Services is greatfully
39 * acknowledged.
40 *
41 * The algorithm remains, but the code has been re-arranged to facilitate
42 * porting to other precisions.
43 */
44
45 /* @(#)gamma.c 8.1 (Berkeley) 6/4/93 */
46 #include <sys/cdefs.h>
47 __FBSDID("$FreeBSD$");
48
49 #include <float.h>
50
51 #include "math.h"
52 #include "math_private.h"
53
54 /* Used in b_log.c and below. */
55 struct Double {
56 double a;
57 double b;
58 };
59
60 #include "b_log.c"
61 #include "b_exp.c"
62
63 /*
64 * The range is broken into several subranges. Each is handled by its
65 * helper functions.
66 *
67 * x >= 6.0: large_gam(x)
68 * 6.0 > x >= xleft: small_gam(x) where xleft = 1 + left + x0.
69 * xleft > x > iota: smaller_gam(x) where iota = 1e-17.
70 * iota > x > -itoa: Handle x near 0.
71 * -iota > x : neg_gam
72 *
73 * Special values:
74 * -Inf: return NaN and raise invalid;
75 * negative integer: return NaN and raise invalid;
76 * other x ~< 177.79: return +-0 and raise underflow;
77 * +-0: return +-Inf and raise divide-by-zero;
78 * finite x ~> 171.63: return +Inf and raise overflow;
79 * +Inf: return +Inf;
80 * NaN: return NaN.
81 *
82 * Accuracy: tgamma(x) is accurate to within
83 * x > 0: error provably < 0.9ulp.
84 * Maximum observed in 1,000,000 trials was .87ulp.
85 * x < 0:
86 * Maximum observed error < 4ulp in 1,000,000 trials.
87 */
88
89 /*
90 * Constants for large x approximation (x in [6, Inf])
91 * (Accurate to 2.8*10^-19 absolute)
92 */
93
94 static const double zero = 0.;
95 static const volatile double tiny = 1e-300;
96 /*
97 * x >= 6
98 *
99 * Use the asymptotic approximation (Stirling's formula) adjusted fof
100 * equal-ripples:
101 *
102 * log(G(x)) ~= (x-0.5)*(log(x)-1) + 0.5(log(2*pi)-1) + 1/x*P(1/(x*x))
103 *
104 * Keep extra precision in multiplying (x-.5)(log(x)-1), to avoid
105 * premature round-off.
106 *
107 * Accurate to max(ulp(1/128) absolute, 2^-66 relative) error.
108 */
109 static const double
110 ln2pi_hi = 0.41894531250000000,
111 ln2pi_lo = -6.7792953272582197e-6,
112 Pa0 = 8.3333333333333329e-02, /* 0x3fb55555, 0x55555555 */
113 Pa1 = -2.7777777777735404e-03, /* 0xbf66c16c, 0x16c145ec */
114 Pa2 = 7.9365079044114095e-04, /* 0x3f4a01a0, 0x183de82d */
115 Pa3 = -5.9523715464225254e-04, /* 0xbf438136, 0x0e681f62 */
116 Pa4 = 8.4161391899445698e-04, /* 0x3f4b93f8, 0x21042a13 */
117 Pa5 = -1.9065246069191080e-03, /* 0xbf5f3c8b, 0x357cb64e */
118 Pa6 = 5.9047708485785158e-03, /* 0x3f782f99, 0xdaf5d65f */
119 Pa7 = -1.6484018705183290e-02; /* 0xbf90e12f, 0xc4fb4df0 */
120
121 static struct Double
large_gam(double x)122 large_gam(double x)
123 {
124 double p, z, thi, tlo, xhi, xlo;
125 struct Double u;
126
127 z = 1 / (x * x);
128 p = Pa0 + z * (Pa1 + z * (Pa2 + z * (Pa3 + z * (Pa4 + z * (Pa5 +
129 z * (Pa6 + z * Pa7))))));
130 p = p / x;
131
132 u = __log__D(x);
133 u.a -= 1;
134
135 /* Split (x - 0.5) in high and low parts. */
136 x -= 0.5;
137 xhi = (float)x;
138 xlo = x - xhi;
139
140 /* Compute t = (x-.5)*(log(x)-1) in extra precision. */
141 thi = xhi * u.a;
142 tlo = xlo * u.a + x * u.b;
143
144 /* Compute thi + tlo + ln2pi_hi + ln2pi_lo + p. */
145 tlo += ln2pi_lo;
146 tlo += p;
147 u.a = ln2pi_hi + tlo;
148 u.a += thi;
149 u.b = thi - u.a;
150 u.b += ln2pi_hi;
151 u.b += tlo;
152 return (u);
153 }
154 /*
155 * Rational approximation, A0 + x * x * P(x) / Q(x), on the interval
156 * [1.066.., 2.066..] accurate to 4.25e-19.
157 *
158 * Returns r.a + r.b = a0 + (z + c)^2 * p / q, with r.a truncated.
159 */
160 static const double
161 #if 0
162 a0_hi = 8.8560319441088875e-1,
163 a0_lo = -4.9964270364690197e-17,
164 #else
165 a0_hi = 8.8560319441088875e-01, /* 0x3fec56dc, 0x82a74aef */
166 a0_lo = -4.9642368725563397e-17, /* 0xbc8c9deb, 0xaa64afc3 */
167 #endif
168 P0 = 6.2138957182182086e-1,
169 P1 = 2.6575719865153347e-1,
170 P2 = 5.5385944642991746e-3,
171 P3 = 1.3845669830409657e-3,
172 P4 = 2.4065995003271137e-3,
173 Q0 = 1.4501953125000000e+0,
174 Q1 = 1.0625852194801617e+0,
175 Q2 = -2.0747456194385994e-1,
176 Q3 = -1.4673413178200542e-1,
177 Q4 = 3.0787817615617552e-2,
178 Q5 = 5.1244934798066622e-3,
179 Q6 = -1.7601274143166700e-3,
180 Q7 = 9.3502102357378894e-5,
181 Q8 = 6.1327550747244396e-6;
182
183 static struct Double
ratfun_gam(double z,double c)184 ratfun_gam(double z, double c)
185 {
186 double p, q, thi, tlo;
187 struct Double r;
188
189 q = Q0 + z * (Q1 + z * (Q2 + z * (Q3 + z * (Q4 + z * (Q5 +
190 z * (Q6 + z * (Q7 + z * Q8)))))));
191 p = P0 + z * (P1 + z * (P2 + z * (P3 + z * P4)));
192 p = p / q;
193
194 /* Split z into high and low parts. */
195 thi = (float)z;
196 tlo = (z - thi) + c;
197 tlo *= (thi + z);
198
199 /* Split (z+c)^2 into high and low parts. */
200 thi *= thi;
201 q = thi;
202 thi = (float)thi;
203 tlo += (q - thi);
204
205 /* Split p/q into high and low parts. */
206 r.a = (float)p;
207 r.b = p - r.a;
208
209 tlo = tlo * p + thi * r.b + a0_lo;
210 thi *= r.a; /* t = (z+c)^2*(P/Q) */
211 r.a = (float)(thi + a0_hi);
212 r.b = ((a0_hi - r.a) + thi) + tlo;
213 return (r); /* r = a0 + t */
214 }
215 /*
216 * x < 6
217 *
218 * Use argument reduction G(x+1) = xG(x) to reach the range [1.066124,
219 * 2.066124]. Use a rational approximation centered at the minimum
220 * (x0+1) to ensure monotonicity.
221 *
222 * Good to < 1 ulp. (provably .90 ulp; .87 ulp on 1,000,000 runs.)
223 * It also has correct monotonicity.
224 */
225 static const double
226 left = -0.3955078125, /* left boundary for rat. approx */
227 x0 = 4.6163214496836236e-1; /* xmin - 1 */
228
229 static double
small_gam(double x)230 small_gam(double x)
231 {
232 double t, y, ym1;
233 struct Double yy, r;
234
235 y = x - 1;
236 if (y <= 1 + (left + x0)) {
237 yy = ratfun_gam(y - x0, 0);
238 return (yy.a + yy.b);
239 }
240
241 r.a = (float)y;
242 yy.a = r.a - 1;
243 y = y - 1 ;
244 r.b = yy.b = y - yy.a;
245
246 /* Argument reduction: G(x+1) = x*G(x) */
247 for (ym1 = y - 1; ym1 > left + x0; y = ym1--, yy.a--) {
248 t = r.a * yy.a;
249 r.b = r.a * yy.b + y * r.b;
250 r.a = (float)t;
251 r.b += (t - r.a);
252 }
253
254 /* Return r*tgamma(y). */
255 yy = ratfun_gam(y - x0, 0);
256 y = r.b * (yy.a + yy.b) + r.a * yy.b;
257 y += yy.a * r.a;
258 return (y);
259 }
260 /*
261 * Good on (0, 1+x0+left]. Accurate to 1 ulp.
262 */
263 static double
smaller_gam(double x)264 smaller_gam(double x)
265 {
266 double d, rhi, rlo, t, xhi, xlo;
267 struct Double r;
268
269 if (x < x0 + left) {
270 t = (float)x;
271 d = (t + x) * (x - t);
272 t *= t;
273 xhi = (float)(t + x);
274 xlo = x - xhi;
275 xlo += t;
276 xlo += d;
277 t = 1 - x0;
278 t += x;
279 d = 1 - x0;
280 d -= t;
281 d += x;
282 x = xhi + xlo;
283 } else {
284 xhi = (float)x;
285 xlo = x - xhi;
286 t = x - x0;
287 d = - x0 - t;
288 d += x;
289 }
290
291 r = ratfun_gam(t, d);
292 d = (float)(r.a / x);
293 r.a -= d * xhi;
294 r.a -= d * xlo;
295 r.a += r.b;
296
297 return (d + r.a / x);
298 }
299 /*
300 * x < 0
301 *
302 * Use reflection formula, G(x) = pi/(sin(pi*x)*x*G(x)).
303 * At negative integers, return NaN and raise invalid.
304 */
305 static double
neg_gam(double x)306 neg_gam(double x)
307 {
308 int sgn = 1;
309 struct Double lg, lsine;
310 double y, z;
311
312 y = ceil(x);
313 if (y == x) /* Negative integer. */
314 return ((x - x) / zero);
315
316 z = y - x;
317 if (z > 0.5)
318 z = 1 - z;
319
320 y = y / 2;
321 if (y == ceil(y))
322 sgn = -1;
323
324 if (z < 0.25)
325 z = sinpi(z);
326 else
327 z = cospi(0.5 - z);
328
329 /* Special case: G(1-x) = Inf; G(x) may be nonzero. */
330 if (x < -170) {
331
332 if (x < -190)
333 return (sgn * tiny * tiny);
334
335 y = 1 - x; /* exact: 128 < |x| < 255 */
336 lg = large_gam(y);
337 lsine = __log__D(M_PI / z); /* = TRUNC(log(u)) + small */
338 lg.a -= lsine.a; /* exact (opposite signs) */
339 lg.b -= lsine.b;
340 y = -(lg.a + lg.b);
341 z = (y + lg.a) + lg.b;
342 y = __exp__D(y, z);
343 if (sgn < 0) y = -y;
344 return (y);
345 }
346
347 y = 1 - x;
348 if (1 - y == x)
349 y = tgamma(y);
350 else /* 1-x is inexact */
351 y = - x * tgamma(-x);
352
353 if (sgn < 0) y = -y;
354 return (M_PI / (y * z));
355 }
356 /*
357 * xmax comes from lgamma(xmax) - emax * log(2) = 0.
358 * static const float xmax = 35.040095f
359 * static const double xmax = 171.624376956302725;
360 * ld80: LD80C(0xdb718c066b352e20, 10, 1.75554834290446291689e+03L),
361 * ld128: 1.75554834290446291700388921607020320e+03L,
362 *
363 * iota is a sloppy threshold to isolate x = 0.
364 */
365 static const double xmax = 171.624376956302725;
366 static const double iota = 0x1p-56;
367
368 double
tgamma(double x)369 tgamma(double x)
370 {
371 struct Double u;
372
373 if (x >= 6) {
374 if (x > xmax)
375 return (x / zero);
376 u = large_gam(x);
377 return (__exp__D(u.a, u.b));
378 }
379
380 if (x >= 1 + left + x0)
381 return (small_gam(x));
382
383 if (x > iota)
384 return (smaller_gam(x));
385
386 if (x > -iota) {
387 if (x != 0.)
388 u.a = 1 - tiny; /* raise inexact */
389 return (1 / x);
390 }
391
392 if (!isfinite(x))
393 return (x - x); /* x is NaN or -Inf */
394
395 return (neg_gam(x));
396 }
397
398 #if (LDBL_MANT_DIG == 53)
399 __weak_reference(tgamma, tgammal);
400 #endif
401